Senate Majority Leader Harry M. Reid (D-Nev.) announced a deal late Tuesday that would allow votes on the legislation, but each of the amendments will need at least 60 votes to be added to the main gun bill.

Senators will consider a bipartisan proposal to expand background checks to cover most commercial gun sales, a plan that lacks sufficient support to pass. They also will vote on Democratic plans to ban military-style assault weapons and limit the size of ammunition magazines; Democratic aides expect that both proposals will not pass.

Votes also will be held on a bipartisan amendment that would make minor changes to the bill’s provisions regarding gun trafficking and a bipartisan plan to provide more funding for mental health programs.

Republican senators secured votes on four proposals. The first, by Charles E. Grassley (Iowa) and Ted Cruz (Tex.), is a package of GOP proposals that would improve the mental health classifications used in the background check reporting system. It is expected to include a proposal Grassley demanded that would prevent abuses like those reported in Operation Fast and Furious, a botched effort to track illegal weapons activity along the U.S.-Mexican border.

Another GOP amendment from Sen. John Cornyn (Tex.) would allow gun owners who receive a state-issued permit to carry a concealed weapon to take that weapon into other states that issue such permits. Only Illinois and the District of Columbia do not issue concealed-carry permits.

What supporters call “national reciprocity” for concealed-carry permits is opposed by gun control groups, which say that adding the proposal to the gun bill would kill the measure, because liberal Democrats would vote against final passage en masse. But establishing reciprocity for state-issued concealed-carry permits is popular in the closely divided Senate, which in recent years has approved plans to carry weapons on Amtrak trains and in national parks that cross state borders.

Supporters of a bipartisan background check plan were struggling Tuesday to find enough support as Senate leaders neared a deal to allow votes on other proposed changes to gun legislation.

The background check proposal, sponsored by Sens. Joe Manchin III (D-W.Va.) and Patrick J. Toomey (R-Pa.), has broad public support but will need at least 60 votes to survive Senate procedural rules if it is to be part of a broader gun control bill. Just 52 senators said they will vote for the plan, according to public statements by senators or their staffs, while 40 senators plan to vote no and seven are undecided.

Despite the worsening odds, Senate Democrats remained confident that the measure will pass with some GOP support.

“This isn’t a 51-49 deal; 90 percent of the American people” support expanding background checks, Reid told reporters. Support for an expanded program dropped slightly, to 86 percent, in a new Washington Post-ABC News poll released Tuesday, but Reid called those numbers impressive.

In hopes of scoring more support for their plan, Manchin and Toomey met with former congresswoman Gabrielle Giffords (D-Ariz.) and her husband, Mark Kelly, on Tuesday. The couple is leading a group seeking new gun laws that was instrumental in bringing together Manchin and Toomey to reach a deal.

Before meeting with Manchin and Toomey, Giffords, who was shot in the head in 2011 in Tucson, told reporters that she is feeling “pretty good.” Later, she and Kelly gave a brief presentation to the weekly Senate Democratic caucus lunch and drew applause that could be heard from outside the closed-door meeting. A few minutes after entering, the couple left and tried to remain optimistic about the legislation’s chances, placing blame not on the contents of the bill but the way others are interpreting it.

“I don’t think it has so many problems. It’s been a short period of time, some folks need to read the legislation,” Kelly said, emphasizing the word “read.”

During the lunch, Manchin delivered an emotional pitch to his colleagues and had tears in his eyes as he cited the recent lobbying work by families of the victims of the Sandy Hook Elementary School massacre in Newtown, Conn., aides said.

The fate of the Manchin-Toomey agreement rests with several undecided senators, including John McCain (R-Ariz.), who said Tuesday that he is inclined to support the proposal but wants more information on how it would regulate background checks for Internet gun sales. The plan’s supporters consider McCain’s backing critical because of his ability to provide political cover for other wavering senators in both parties.

In hopes of dampening GOP support for the proposal, Grassley raised doubts about the background check plan, saying that significant disagreement remains. “We haven’t voted on it because supporters don’t have the votes to pass it, at least at this point,” he said.

Sen. Richard J. Durbin (Ill.), the Democratic whip, said there is no fallback plan for crafting modified background check legislation if the current draft loses on the Senate floor. “I think Manchin-Toomey is our best hope of passing a background check,” he said.

Ahead of the votes, the National Rifle Association issued fresh warnings to its millions of members Tuesday, saying in a fundraising e-mail that President Obama and Senate Democrats are “fighting to BAN tens of millions of commonly owned firearms . . . fighting to REGISTER and LICENSE gun owners . . . fighting to create a FEDERAL REGISTRY of ammo buyers . . . and fighting to DESTROY your right to defend yourself, your home and your loved ones.”